Finite Group Databasebeta

← #11824440

Subgroups — #11824440

Order 1152.

2109 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64090C211yesmaximal, characteristic, complemented
2C2×#63980C212yesmaximal, complemented
3C2×#63979C212yesmaximal, complemented
4C2×#63914C211yesmaximal, characteristic, complemented
5C2×#63912C212yesmaximal, complemented
6C3×#4960C212yesmaximal, complemented
7#10591391C212yesmaximal, complemented
8#10591386C212yesmaximal, complemented
9#10591257C211yesmaximal, characteristic, complemented
10#76688—32nomaximal
11C22×#638C2211yescharacteristic
12C23×C3⋊Dic3C2211yescharacteristic
13C22×C4×C3⋊D3C2211yescharacteristic
14C22×C3×#67C2214yes—
15C22×C3×D12C2212yes—
16C22×#626C2214yes—
17C22×#625C2212yes—
18C22×#625C2212yes—
19C2×C32×C22⋊C4C2211yescharacteristic
20C2×#3752C2211yescharacteristic
21C2×#3750C2211yescharacteristic
22C2×C4×C3⋊Dic3C2211yescharacteristic
23#64090—24no—
24C2×C3×(C2×D3)⋊C4C2214yes—
25#63980—28no—
26#63979—28no—
27C2×#3724C2212yes—
28C2×(C2×D3)⋊Dic3C2214yes—
29C2×Dic32C2212yes—
30#63914—24no—
31#63912—28no—
32C3×C3⋊#95C2212yes—
33C2×#940—34no—
34C2×#940—32no—
35C2×#939—34no—
36C2×#884—34no—
37C2×#67⋊C4—32no—
38C2×#67⋊C4—61no—
39C3×#392—32no—
40#5223—32no—
41#4960D312yescomplemented
42#4957—34no—
43#4671—32no—
44#4649—32no—
45C24×C32C2311yescharacteristic
46C23×C3⋊D3C2311yescharacteristic
47C23×C3×D3C2314yes—
48C22×C32×C4C2311yescharacteristic
49C22×C32×C4C2311yescharacteristic
50C2×#638—22no—
51C2×#638—22no—
52C2×#638D414yescomplemented
53C22×C3⋊Dic3—22no—
54C22×C3⋊Dic3—24no—
55C22×C3⋊Dic3C2311yescharacteristic
56C22×C3⋊Dic3C2311yescharacteristic
57C22×C3⋊Dic3C2311yescharacteristic
58C2×C4×C3⋊D3—22no—
59C2×C4×C3⋊D3—24no—
60C2×C3×#67—28no—
61C2×C3×#67—28no—
62C2×C3×#67—44no—
63C22×C3×Dic3C2314yes—
64C2×C3×D12—24no—
65C2×C3×D12—44no—
66C2×#626—28no—
67C2×#626—28no—
68C2×#626—44no—
69C2×#625—24no—
70C2×#625—24no—
71C2×#625—24no—
72C2×#625—24no—
73C2×#625—24no—
74C2×#625—24no—
75C32×C22⋊C4—22no—
76#3752—22no—
77#3750—22no—
78C4×C3⋊Dic3D414yescomplemented
79C3×(C2×D3)⋊C4—28no—
80#3724D418yescomplemented
81(C2×D3)⋊Dic3—28no—
82Dic32—24no—
83#2541—91noSylow
84C22×C3×D4—34no—
85C22×C3×D4—32no—
86C22×#67—32no—
87C22×#67—34no—
88C22×#67—34no—
89C22×#67—34no—
90C22×#67—61no—
91C22×#67C2×D314yes—
92C23×Dic3—32no—
93C23×Dic3—61no—
94C22×D12—34no—
95C22×D12C2×D312yes—
96C22×C4×D3—32no—
97C22×C4×D3—61no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—34no—
100C2×C3×C22⋊C4C2×D312yes—
101C2×#217—34no—
102#940—68no—
103#940—616no—
104#939—616no—
105C2×(C2×D3)⋊C4—34no—
106C2×(C2×D3)⋊C4—32no—
107C2×(C2×D3)⋊C4—61no—
108C2×(C2×D3)⋊C4C2×D314yes—
109C2×Dic3⋊C4—34no—
110C2×Dic3⋊C4—32no—
111C2×Dic3⋊C4—61no—
112C2×C4×Dic3—32no—
113C2×C4×Dic3—34no—
114C2×C4×Dic3—61no—
115#884—616no—
116#67⋊C4—68no—
117#67⋊C4—68no—
118C3×#95—32no—
119C3⋊#95C2×D312yes—
120C23×C32—21no—
121C23×C32—21no—
122C23×C32—22no—
123C23×C32—22no—
124C23×C32C2×D412yes—
125C23×C32C2411yescharacteristic
126C22×C3⋊D3—21no—
127C22×C3⋊D3—21no—
128C22×C3⋊D3—22no—
129C22×C3⋊D3—22no—
130C22×C3⋊D3C2×D412yes—
131C22×C3×D3—24no—
132C22×C3×D3—24no—
133C22×C3×D3—24no—
134C22×C3×D3—24no—
135C22×C3×D3—24no—
136C22×C3×D3—24no—
137C22×C3×D3—24no—
138C2×C32×C4—21no—
139C2×C32×C4—22no—
140C2×C32×C4—22no—
141C2×C32×C4C2×D412yes—
142#638—28no—
143C2×C3⋊Dic3—21no—
144C2×C3⋊Dic3—22no—
145C2×C3⋊Dic3—44no—
146C2×C3⋊Dic3C2×D412yes—
147C2×C3⋊Dic3C2×D412yes—
148C2×C3⋊Dic3C2×D414yes—
149C2×C3⋊Dic3C2×D412yes—
150C2×C3⋊Dic3#4112yes—
151C4×C3⋊D3—44no—
152C3×#67—416no—
153C2×C3×Dic3—24no—
154C2×C3×Dic3—24no—
155C2×C3×Dic3C2×D418yes—
156C3×D12—48no—
157#626—416no—
158#625—48no—
159#625—48no—
160C2×#127—91no—
161C2×#127—92no—
162C2×#121—92no—
163C2×#121—92no—
164C2×C4×D4—91no—
165#392—32no—
166#392—92no—
167#390—92no—
168C4×C4⋊C4—91no—
169C24×C3—21no—
170C24×C3—34no—
171C24×C3C22×D312yes—
172C23×D3—32no—
173C23×D3—61no—
174C23×D3C22×D314yes—
175C2×C3×D4—68no—
176C2×C3×D4—68no—
177C2×C3×D4—64no—
178C2×C3×D4—124no—
179C2×C3×D4—124no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—34no—
183C22×C3×C4C22×D312yes—
184C22×C3×C4C22×D312yes—
185C2×#67—28no—
186C2×#67—28no—
187C2×#67—38no—
188C2×#67—44no—
189C2×#67—64no—
190C2×#67—68no—
191C2×#67—64no—
192C2×#67—64no—
193C2×#67—68no—
194C2×#67—68no—
195C2×#67—68no—
196C2×#67—68no—
197C2×#67—68no—
198C2×#67—68no—
199C2×#67—64no—
200C2×#67—64no—
201C2×#67—68no—
202C2×#67—124no—
203C22×Dic3—32no—
204C22×Dic3—32no—
205C22×Dic3—32no—
206C22×Dic3—61no—
207C22×Dic3—61no—
208C22×Dic3—61no—
209C22×Dic3—64no—
210C22×Dic3—68no—
211C22×Dic3—64no—
212C22×Dic3—124no—
213C22×Dic3C22×D314yes—
214C2×D12—24no—
215C2×D12—44no—
216C2×D12—68no—
217C2×D12—68no—
218C2×D12—124no—
219C2×C4×D3—64no—
220C2×C4×D3—68no—
221C2×C4×D3—64no—
222C2×C4×D3—124no—
223C3×C22⋊C4—24no—
224C3×C22⋊C4—24no—
225C3×C22⋊C4—68no—
226#217—68no—
227(C2×D3)⋊C4—28no—
228(C2×D3)⋊C4—64no—
229(C2×D3)⋊C4—68no—
230(C2×D3)⋊C4—64no—
231Dic3⋊C4—316no—
232Dic3⋊C4—64no—
233Dic3⋊C4—64no—
234C4×Dic3—38no—
235C4×Dic3—64no—
236C4×Dic3—68no—
237C22×C32—22no—
238C22×C32—24no—
239C22×C32—41no—
240C22×C32—41no—
241C22×C32—42no—
242C22×C32C22×D411yescharacteristic
243C22×C32C22×D411yescharacteristic
244C22×C32C22×D412yes—
245C22×C32C2×#4111yescharacteristic
246C22×C32H(5,𝔽2)12yes—
247C2×C3⋊D3—22no—
248C2×C3⋊D3—24no—
249C2×C3⋊D3—41no—
250C2×C3⋊D3—41no—
251C2×C3⋊D3—42no—
252C2×C3×D3—44no—
253C2×C3×D3—44no—
254C2×C3×D3—44no—
255C2×C3×D3—44no—
256C2×C3×D3—44no—
257C2×C3×D3—44no—
258C2×C3×D3—44no—
259C32×C4—22no—
260C32×C4—41no—
261C3⋊Dic3—41no—
262C3⋊Dic3#12114yescomplemented
263C3⋊Dic3#12714yescomplemented
264C3×Dic3—28no—
265C22×D4—34no—
266C22×D4—32no—
267C22×D4—94no—
268C22×D4—91no—
269C22×D4—92no—
270C22×D4—92no—
271C23×C4—91no—
272C23×C4—91no—
273#127—184no—
274#127—188no—
275#121—188no—
276#121—188no—
277C4×D4—184no—
278C2×C4⋊C4—92no—
279C2×C4⋊C4—91no—
280C2×C22⋊C4—34no—
281C2×C22⋊C4—94no—
282C2×C22⋊C4—91no—
283C2×C22⋊C4D3211yescharacteristic
284C2×C42—91no—
285C2×C42—92no—
286#95—32no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—24no—
291C23×C3—24no—
292C23×C3—22no—
293C23×C3—21no—
294C23×C3—22no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—64no—
298C23×C3—64no—
299C23×C3—64no—
300C23×C3—64no—
301C23×C3—64no—
302C23×C3—64no—
303C23×C3—64no—
304C23×C3D3×D414yes—
305C23×C3C23×D312yes—
306C22×D3—24no—
307C22×D3—24no—
308C22×D3—24no—
309C22×D3—24no—
310C22×D3—24no—
311C22×D3—24no—
312C22×D3—24no—
313C22×D3—34no—
314C22×D3—62no—
315C22×D3—62no—
316C22×D3—62no—
317C22×D3—62no—
318C22×D3—64no—
319C22×D3—64no—
320C22×D3—62no—
321C22×D3—122no—
322C22×D3—122no—
323C3×D4—128no—
324C3×D4—1216no—
325C2×C3×C4—24no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—24no—
329C2×C3×C4—22no—
330C2×C3×C4—38no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—64no—
334C2×C3×C4—64no—
335C2×C3×C4D3×D414yes—
336#67—416no—
337#67—616no—
338#67—616no—
339#67—1216no—
340#67—1216no—
341#67—1216no—
342C2×Dic3—24no—
343C2×Dic3—24no—
344C2×Dic3—34no—
345C2×Dic3—34no—
346C2×Dic3—34no—
347C2×Dic3—38no—
348C2×Dic3—34no—
349C2×Dic3—62no—
350C2×Dic3—62no—
351C2×Dic3—64no—
352C2×Dic3—62no—
353C2×Dic3—62no—
354C2×Dic3—62no—
355C2×Dic3—62no—
356C2×Dic3—64no—
357C2×Dic3—122no—
358C2×Dic3—128no—
359C2×Dic3—128no—
360C2×Dic3D3×D418yes—
361D12—48no—
362D12—1216no—
363C4×D3—128no—
364C4×D3—128no—
365C2×C32—42no—
366C2×C32C2×#12111yescharacteristic
367C2×C32C2×#12711yescharacteristic
368C2×C32#53511yescharacteristic
369C2×C32D4212yes—
370C2×C32#55012yes—
371C3⋊D3—42no—
372C3×D3—84no—
373C24—34no—
374C24—91no—
375C24C2×D3211yescharacteristic
376C2×D4—68no—
377C2×D4—68no—
378C2×D4—64no—
379C2×D4—94no—
380C2×D4—124no—
381C2×D4—124no—
382C2×D4—188no—
383C2×D4—188no—
384C2×D4—182no—
385C2×D4—182no—
386C2×D4—184no—
387C2×D4—184no—
388C2×D4—184no—
389C2×D4—184no—
390C2×D4—184no—
391C2×D4—184no—
392C2×D4—364no—
393C22×C4—34no—
394C22×C4—91no—
395C22×C4—91no—
396C22×C4—91no—
397C22×C4—182no—
398C22×C4—184no—
399C22×C4—182no—
400C22×C4—184no—
401C22×C4C2×D3211yescharacteristic
402C22×C4C2×D3211yescharacteristic
403C4⋊C4—98no—
404C4⋊C4—182no—
405C22⋊C4—22no—
406C22⋊C4—68no—
407C22⋊C4—188no—
408C22⋊C4—182no—
409C42—94no—
410C42—184no—
411C22×C3—24no—
412C22×C3—28no—
413C22×C3—24no—
414C22×C3—21no—
415C22×C3—21no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—21no—
419C22×C3—44no—
420C22×C3—42no—
421C22×C3—44no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—44no—
426C22×C3—124no—
427C22×C3—124no—
428C22×C3—124no—
429C22×C3—124no—
430C22×C3—124no—
431C22×C3—124no—
432C22×C3—124no—
433C22×C3C2×D3×D412yes—
434C22×C3C2×D3×D414yes—
435C22×C3C2×D3×D412yes—
436C22×C3#100414yes—
437C22×C3D3×#4112yes—
438C2×D3—44no—
439C2×D3—44no—
440C2×D3—44no—
441C2×D3—44no—
442C2×D3—44no—
443C2×D3—44no—
444C2×D3—44no—
445C2×D3—64no—
446C2×D3—68no—
447C2×D3—64no—
448C2×D3—124no—
449C2×D3—122no—
450C2×D3—124no—
451C2×D3—122no—
452C2×D3—122no—
453C2×D3—122no—
454C2×D3—124no—
455C3×C4—24no—
456C3×C4—42no—
457C3×C4—42no—
458C3×C4—42no—
459C3×C4—68no—
460Dic3—28no—
461Dic3—38no—
462Dic3—38no—
463Dic3—64no—
464Dic3—64no—
465Dic3—122no—
466Dic3—122no—
467C32#254111yescharacteristic, complemented, Sylow
468C23—21no—
469C23—22no—
470C23—22no—
471C23—21no—
472C23—64no—
473C23—64no—
474C23—64no—
475C23—64no—
476C23—64no—
477C23—64no—
478C23—64no—
479C23—92no—
480C23—181no—
481C23—182no—
482C23—182no—
483C23—181no—
484C23#380212yes—
485C23C22×D3211yescharacteristic
486D4—128no—
487D4—1216no—
488D4—188no—
489D4—3616no—
490D4—368no—
491D4—368no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—38no—
496C2×C4—64no—
497C2×C4—64no—
498C2×C4—92no—
499C2×C4—92no—
500C2×C4—94no—
501C2×C4—92no—
502C2×C4—92no—
503C2×C4—182no—
504C2×C4—181no—
505C2×C4—364no—
506C2×C4—364no—
507C2×C4#381112yes—
508C2×C3—21no—
509C2×C3—22no—
510C2×C3—21no—
511C2×C3—22no—
512C2×C3—21no—
513C2×C3—44no—
514C2×C3—44no—
515C2×C3—244no—
516C2×C3#559014yes—
517C2×C3#559212yes—
518C2×C3#559914yes—
519C2×C3D3×#12112yes—
520C2×C3D3×#12712yes—
521D3—84no—
522D3—124no—
523D3—124no—
524C22—22no—
525C22—24no—
526C22—42no—
527C22—41no—
528C22—41no—
529C22—124no—
530C22—124no—
531C22—124no—
532C22—124no—
533C22—124no—
534C22—124no—
535C22—124no—
536C22—182no—
537C22—184no—
538C22—362no—
539C22—361no—
540C22—361no—
541C22C2×#380211yescharacteristic
542C22#6430611yescharacteristic
543C22D32×D412yes—
544C22#6431312yes—
545C22C2×#381111yescharacteristic
546C4—22no—
547C4—41no—
548C4—68no—
549C4—94no—
550C4—94no—
551C4—361no—
552C3—21no—
553C3#7668812yescomplemented
554C2—42no—
555C2—244no—
556C2—362no—
557C2#1059419012yes—
558C2#1059427112yes—
559C2#1059508111yescharacteristic
560C2#1059523911yescharacteristic
561C2#1059530011yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutator, socle, cosocleC23×C32C24
FittingC2×C32×C22⋊C4C22
hypercenterC2×C22⋊C4D32
hypocenterC32#2541
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#64090C2semidirect
2. C2×#63980C2semidirect
3. C2×#63979C2semidirect
4. C2×#63914C2semidirect
5. C2×#63912C2semidirect
6. C3×#4960C2semidirect
7. #10591391C2semidirect
8. #10591386C2semidirect
9. #10591257C2semidirect
41. #4960D3semidirect
52. C2×#638D4semidirect
78. C4×C3⋊Dic3D4semidirect
80. #3724D4semidirect
262. C3⋊Dic3#121semidirect
263. C3⋊Dic3#127semidirect
467. C32#2541semidirect
553. C3#76688semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#640902
C2×#639802
C2×#639792
C2×#639142
C2×#639122
C3×#49602
#105913912
#105913862
#105912572
#766883